Activity – sets of 3 r th term r+1 th term first few terms

Sequences and Series Use subscript notation Use subscript notation Generate a sequence using a rule Generate a sequence using a rule Generate a sequence using inductive definition Generate a sequence using inductive definition Describing the behaviour of a sequence Describing the behaviour of a sequence

Sequences FiniteInfinite , 4, 7, 10… A finite sequence has a first and last term A infinite sequence goes on forever

Notation If we say u n = 3n - 5 u 1 = 3(1) - 5 u 2 = 3(2) - 5 u 3 = 3(3) - 5 u 4 = 3(4) - 5 u 5 = 3(5) - 5 = -2 = 1 = 4 = 7 = 10

Sequences For each of the following, work out the first 6 terms, the 50 th term and the 100 th term. Describe what happens as n gets larger. u n = 3n + 2 c n = 7 – 4n k n = n 2 – 2n + 8 r n = (-2) n

Inductive Definition u 1 = 3 u 2 = 8 u 3 = 13 u 4 = 18 Write down the terms of the sequence u n = 5n - 2

Inductive Definition u 1 = 3 u 2 = 8 u 3 = 13 u 4 = 18 Each term is 5 more than the last u 2 = u u 3 = u u 4 = u We could write this as: u n+1 = u n + 5 Next termCurrent term

Inductive Definition This is called a recurrence relation u n+1 = u n + 5

Inductive Definition Write down a sequence using the recurrence relation above. u n+1 = u n - 3 What extra information do I need?

Inductive Definition In order to write a sequence when given a recurrence relation, I need to know the first term. u n+1 = u n - 3

Inductive Definition Now write the sequence u n+1 = u n - 3u 1 = 7 u 1 = 7 u 2 = 4 u 3 = 1 u 4 = -2 u 5 = -5

Problem A sequence is defined by u 1 = 3u n+1 = u n + k If u 4 = 7.5, find k u 1 = 3 u 2 = 3 + k u 3 = 3 + k + k = 3 + 2k u 4 = 3 + 2k + k = 3 + 3k 3 + 3k = 7.5k = 1.5 U n+1 = u n + 1.5

Quiz